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Chapter 6: Problem 45

Simplify each expression. $$\frac{\tan x}{3}+\frac{\tan x}{2}$$

Short Answer

Expert verified
\frac{5 \tan x }{6}

Step by step solution

01

Identify Common Terms

Both fractions share the common term \( \tan x \). Combining these will simplify the expression.
02

Find a Common Denominator

Add the fractions by finding a common denominator. The common denominator for 3 and 2 is 6.
03

Rewrite Fractions with Common Denominator

Rewrite \(\frac{\tan x}{3}\) and \(\frac{\tan x}{2}\) with a denominator of 6: \[\frac{\tan x}{3} = \frac{2\tan x}{6}\] and \[\frac{\tan x}{2} = \frac{3\tan x}{6}\]
04

Combine the Fractions

Add the fractions with the same denominator: \[\frac{2\tan x}{6} + \frac{3\tan x}{6} = \frac{(2 + 3)\tan x}{6} = \frac{5\tan x}{6}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Functions
Trigonometric functions play a crucial role in mathematics, especially when dealing with angles and periodic phenomena. Functions like sine, cosine, and tangent are foundational. In this exercise, we focus on the tangent function, denoted as \(\tan x\). It is defined as the ratio of the opposite side to the adjacent side in a right triangle. The expression we simplify involves fractions of the tangent function. Understanding this function helps in combining and simplifying expressions with ease.
Common Denominator
Finding a common denominator is essential when adding fractions together. It involves finding a number that multiples of both denominators share. In this exercise, we add the fractions \(\frac{\tan x}{3}\) and \(\frac{\tan x}{2}\).
First, identify the least common multiple (LCM) of the denominators 3 and 2, which is 6. This common denominator allows combining the fractions:

  • Rewrite \(\frac{\tan x}{3}\) as \(\frac{2\tan x}{6}\)
  • Rewrite \(\frac{\tan x}{2}\) as \(\frac{3\tan x}{6}\)
With a common denominator, these fractions are now easily added.
Fraction Addition
Adding fractions is straightforward once they share a common denominator. Let's look at our expression:

After rewriting \(\frac{\tan x}{3}\) and \(\frac{\tan x}{2}\) as \(\frac{2\tan x}{6}\) and \(\frac{3\tan x}{6}\), you sum them up. This step-by-step process ensures clarity:

  • Add the numerators: \(\frac{2\tan x}{6} + \frac{3\tan x}{6} = \frac{(2+3)\tan x}{6}\)
  • Combine terms: This results in \(\frac{5\tan x}{6}\).
These steps consolidate what we learned about common denominators and the addition of fractions, culminating in a simplified final expression.

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Most popular questions from this chapter

In each case, find \(\sin \alpha, \cos \alpha, \tan \alpha, \csc \alpha, \sec \alpha,\) and \(\cot \alpha\) $$\sin (\alpha / 2)=-1 / 3 \text { and } 7 \pi / 4<\alpha / 2<2 \pi$$

Find the exact value of \(\sin (x / 2)\) given that \(\cos (x)=1 / 4\) and \(3 \pi / 2

Verify that each equation is an identity. $$\tan ^{2}\left(\frac{x}{2}\right)=\frac{\sec x+\cos x-2}{\sec x-\cos x}$$

Solve each problem. Wave Action The vertical position of a floating ball in an experimental wave tank is given by the equation \(x=2 \sin (\pi t / 3)\) where \(x\) is the number of feet above sea level and \(t\) is the time in seconds. For what values of \(t\) is the ball \(\sqrt{3} \mathrm{ft}\) above sea level?

Find the exact value of \(\sin (x / 2)\) given that \(\cos (x)=-1 / 4\) and \(\pi / 2
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